# Quadratic Equations in Square Roots

For solving Quadratic Equations in Square Roots, it is required to square both sides entirely, but not individually. For instance,
$$(1 + 2)^2 = 3^2 \\ 1^2 + 2^2 \ne 3^2$$
It is important to check the solutions to see they work for the original equation, if the original equation is squared.

### Worked Examples of Quadratic Equations in Square Roots

Solve $\sqrt{x+5} + \sqrt{x-2} = \sqrt{5x-6}$ .

\begin{aligned} \require{color} \displaystyle \Big(\sqrt{x+5} + \sqrt{x-2}\Big)^2 &= \sqrt{5x-6}^2 &\color{red} \text{square both sides} \\ x+5 + 2 \sqrt{x+5} \sqrt{x-2} + x-2 &= 5x-6 \\ 2 \sqrt{x+5} \sqrt{x-2} &= 3x-9 \\ 4(x+5)(x-2) &= (3x-9)^2 &\color{red} \text{square both sides} \\ 4(x^2 + 3x – 10) &= 9x^2 – 54x + 81 \\ 5x^2 -66x + 121 &= 0 \\ (5x-11)(x-11) &=0 \\ x = \frac{11}{5} \text{ or } 11 \\ \end{aligned} \\
$\text{Check whether the answers are OK, but } x= \displaystyle \frac{11}{5} \text{ does not make the original equation.} \\ \text{So the answer is only } x=11. \\$ 